Assume that $n$ is the sum of two square numbers, then it is congruent to $0+0 = 0$, $1+0 = 1$, $0+1 = 1$ or $1+1 = 2$ modulo $4$. The contrapositive tells us that if $n$ is not congruent to either $0$, $1$ or $2$ modulo $4$ then it is not the sum of two square numbers. This proves your statement.